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Understanding Least-Squares Regression in Describing the Relation between Two Variables

Learn how to find and interpret the least-squares regression line, make predictions, and compute the sum of squared residuals.

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Understanding Least-Squares Regression in Describing the Relation between Two Variables

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  1. Chapter 4 Describing the Relation between Two Variables

  2. Section 4.2 Least-squares Regression

  3. Objectives • Find the least-squares regression line and use the line to make predictions • Interpret the slope and the y-intercept of the least-squares regression line • Compute the sum of squared residuals

  4. EXAMPLE Finding an Equation that Describes Linearly Relate Data Using the following sample data: (a) Find a linear equation that relates x (the explanatory variable) and y (the response variable) by selecting two points and finding the equation of the line containing the points. Using (2, 5.7) and (6, 1.9):

  5. (b) Graph the equation on the scatter diagram. (c) Use the equation to predict y if x = 3.

  6. Objective 1 Find the Least-Squares Regression Line and Use the Line to Make Predictions

  7. The difference between the observed value of y and the predicted value of y is the error, or residual. Using the line from the last example, and the predicted value at x = 3: residual = observed y – predicted y = 5.2 – 4.75 = 0.45 (3, 5.2) } residual = observed y – predicted y = 5.2 – 4.75 = 0.45

  8. Least-Squares Regression Criterion The least-squares regression line is the line that minimizes the sum of the squared errors (or residuals). This line minimizes the sum of the squared vertical distance between the observed values of y and those predicted by the line, (“y-hat”). We represent this as “minimize Σ residuals2 ”.

  9. The Least-Squares Regression Line The equation of the least-squares regression line is given by where is the slope of the least-squares regression line where is the y-intercept of the least-squares regression line

  10. The Least-Squares Regression Line Note: is the sample mean and sx is the sample standard deviation of the explanatory variable x ; is the sample mean and sy is the sample standard deviation of the response variable y.

  11. EXAMPLE Finding the Least-squares Regression Line • Using the drilling data • Find the least-squares regression line. • Predict the drilling time if drilling starts at 130 feet. • Is the observed drilling time at 130 feet above, or below, average. • Draw the least-squares regression line on the scatter diagram of the data.

  12. We agree to round the estimates of the slope and intercept to four decimal places. (b) (c) The observed drilling time is 6.93 seconds. The predicted drilling time is 7.035 seconds. The drilling time of 6.93 seconds is below average.

  13. (d)

  14. Objective 2 Interpret the Slope and the y-Intercept of the Least-Squares Regression Line

  15. Interpretation of Slope: The slope of the regression line is 0.0116. For each additional foot of depth we start drilling, the time to drill five feet increases by 0.0116 minutes, on average.

  16. Interpretation of the y-Intercept: The y-intercept of the regression line is 5.5273. To interpret the y-intercept, we must first ask two questions: 1. Is 0 a reasonable value for the explanatory variable? 2. Do any observations near x = 0 exist in the data set? A value of 0 is reasonable for the drilling data (this indicates that drilling begins at the surface of Earth. The smallest observation in the data set is x = 35 feet, which is reasonably close to 0. So, interpretation of the y-intercept is reasonable. The time to drill five feet when we begin drilling at the surface of Earth is 5.5273 minutes.

  17. If the least-squares regression line is used to make predictions based on values of the explanatory variable that are much larger or much smaller than the observed values, we say the researcher is working outside the scope of the model. Never use a least-squares regression line to make predictions outside the scope of the model because we can’t be sure the linear relation continues to exist.

  18. Objective 3 Compute the Sum of Squared Residuals

  19. To illustrate the fact that the sum of squared residuals for a least-squares regression line is less than the sum of squared residuals for any other line, use the “regression by eye” applet.

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